
We expect that the front of the
solution to (REF ) shifts from exhibiting unbounded transition
fronts (essentially when \(\mathsf {es}- \mathsf {ei}\)  large, and maybe further conditions, cf. Theorem REF )
to exhibiting bounded transition fronts (essentially if \(\mathsf {es}- \mathsf {ei}\) 
small, and maybe further conditions, cf. (REF )). While it is not clear if
‘‘small” means ‘‘vanishes” in this context, let us point out here
that—while periodic media are oftentimes taken to be a simple instance
for heterogeneous or random media, cf. also [1]}, [2]}, [3]}—it is clear from our proofs that the
phenomenon of long stretches of areas of high and low potential, which is
crucial in our proof, is not observed for periodic media.

Is there a logarithmic upper bound corresponding to the result of Theorem REF  as well, in the sense that \(m^\varepsilon (t) - m^{1-\varepsilon ,-}(t) \le C \log t\)  for all \(t\)  large enough?

